[F(x):=,S(k,x):=]

M(k,x,h):=INT(S(2*k,(y-x)/h)*DIF(F(y),y,2*k),y,x,x+h)/(2*k)!

M(k,x,h)=-INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)/(h*(2*k-1)!)

[V(y):=,U(y):=]

INT(V(y)*DIF(U(y),y),y,x,x+h)=LIM(V(y)*U(y),y,x+h,0)-LIM(V(y)*U(y),y,x,0)-INT~
(U(y)*DIF(V(y),y),y,x,x+h)

[V(y):=S(2*k-1,(y-x)/h),U(y):=DIF(F(y),y,2*k-2)]

;Simp(#5)
INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)=-INT(DIF(F(y),y,2*k-2)*LIM(DI~
F(S(2*k-1,@),@),@,(y-x)/h,0),y,x,x+h)/h+LIM(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-2~
),y,x+h,0)-LIM(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-2),y,x,0)

[U(y)=DIF(F(y),y,2*k-2),U(y):=]

;Simp(User)
INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)=-INT(U(y)*LIM(DIF(S(2*k-1,@),~
@),@,(y-x)/h,0),y,x,x+h)/h+U(x+h)*S(2*k-1,1)-U(x)*S(2*k-1,0)

[S(k,x):=k*SUM(p^(k-1),p,1,x-1),k:epsilonInteger (1, inf)]

;User=Simp(User)
[S(2*k-1,0),S(2*k-1,1)]=[0,0]

[S(k,x):=,B(k):=]

;Simp(User)
INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)=-INT(U(y)*LIM(DIF(S(2*k-1,@),~
@),@,(y-x)/h,0),y,x,x+h)/h

LIM(DIF(S(2*k-1,@),@),@,(y-x)/h,0)=LIM(DIF(S(2*k-1,t),t),t,(y-x)/h,0)

[DIF(S(k,x),x)=k*(S(k-1,x)+B(k-1)),k>2]

;Simp(User)
DIF(S(2*k-1,t),t)=(2*k-1)*(B(2*k-2)+S(2*k-2,t))

;Simp(User)
LIM(DIF(S(2*k-1,@),@),@,(y-x)/h,0)=(2*k-1)*(B(2*k-2)+S(2*k-2,(y-x)/h))

;Simp(User)
INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)=(1-2*k)*(B(2*k-2)*INT(U(y),y,~
x,x+h)+INT(U(y)*S(2*k-2,(y-x)/h),y,x,x+h))/h

U(y):=DIF(F(y),y,2*k-2)

;Simp(User)
INT(S(2*k-1,(y-x)/h)*DIF(F(y),y,2*k-1),y,x,x+h)=(1-2*k)*(B(2*k-2)*(LIM(DIF(F(~
y),y,2*k-3),y,x+h,0)-LIM(DIF(F(y),y,2*k-3),y,x,0))+INT(S(2*k-2,(y-x)/h)*DIF(F~
(y),y,2*k-2),y,x,x+h))/h

;Simp(User')
M(k,x,h)=(INT(S(2*k-2,(y-x)/h)*DIF(F(y),y,2*k-2),y,x,x+h)-B(2*k-2)*(LIM(DIF(F~
(y),y,2*k-3),y,x+h,0)-LIM(DIF(F(y),y,2*k-3),y,x,0)))/(h^2*(2*k-2)!)

;User=Simp(User)
M(k-1,x,h)=INT(S(2*(k-1),(y-x)/h)*DIF(F(y),y,2*(k-1)),y,x,x+h)/(2*k-2)!

;User=Simp(User)
M(k,x,h):=

M(k-1,x,h)*(2*k-2)!=INT(S(2*(k-1),(y-x)/h)*DIF(F(y),y,2*(k-1)),y,x,x+h)

;Simp(User)
M(k,x,h)=B(2*k-2)*(LIM(DIF(F(y),y,2*k-3),y,x+h,0)-LIM(DIF(F(y),y,2*k-3),y,x,0~
))/(h^2*(2*k-2)!)+M(k-1,x,h)/h^2